TITLE: On a trace condition for positive definite $2\times 2$ block matrices
QUESTION [6 upvotes]: Consider the following block matrix
$$
X = \begin{bmatrix} A & C \\ C^\top & B\end{bmatrix},
$$
where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, and $C\in\mathbb{R}^{n\times m}$.
To Prove (or disprove): If $X$ is positive definite, i.e. $X>0$, then the following trace inequality holds
$$
\left[\mathrm{tr}(CC^\top)\right]^2< \mathrm{tr}(A^2)\mathrm{tr}(B^2).
$$
Some comments. Based on Theorem 2.3 of Horn and Mathias. "Cauchy-Schwarz inequalities associated with positive semidefinite matrices." Linear Algebra and Its Applications 142 (1990): 63-82, I think it is possible to prove the previous fact if we replace strict inequalities with non-strict ones.

REPLY [6 votes]: For any unitarily invariant norm it can be shown that
\begin{equation*}
  \|X\| = \left\Vert 
    \begin{bmatrix}
      A & C\\
      C^* & B
    \end{bmatrix}
  \right\Vert \le \|A\| + \|B\|.
\end{equation*}
Thus, using the squared Frobenius norm on both sides and cancelling, we obtain
\begin{equation*}
  \|C\|_F^2 \le \|A\|_F\|B\|_F,
\end{equation*}
as desired.